# 複素数計算機

> Add, subtract, multiply and divide complex numbers, raise them to powers, find nth roots, and convert a + bi to polar form, with an Argand diagram.

操作できるページ：https://www.calcopenly.com/ja/math/complex-number-calculator
分野：数学の計算機

A complex number z = a + bi has a real part a and an imaginary part b, where i² = −1. Addition, subtraction and multiplication work in rectangular form; division multiplies top and bottom by the conjugate of the divisor. Powers and roots use polar form, z = r(cos θ + i sin θ): De Moivre's formula raises the modulus r to the power n and multiplies the angle θ by n.

Electrical engineers use complex numbers for AC impedance and phasors, and algebra students meet them as roots of polynomials. The default multiplies (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i, whose modulus is √125 ≈ 11.1803.

Inputs can be rectangular (3 + 4i, with j accepted for i) or polar (2∠90). Angles follow the degree, radian or gradian setting, and the argument is given between −180° and 180°. The Argand diagram draws each number as an arrow from the origin.

## 入力

- **Operation** (選択肢：Convert to polar form, Add z₁ + z₂, Subtract z₁ − z₂, Multiply z₁ × z₂, Divide z₁ ÷ z₂, Power z₁ⁿ, nth roots of z₁)
- **z₁**: Rectangular a + bi (j also works) or polar r∠θ; θ uses the angle unit from settings unless you add ° or rad
- **z₂**
- **n**: Power: any real number. Roots: a whole number from 1 to 360.

## 結果

- 結果 — 主な結果
- Real part
- Imaginary part
- Modulus |z|
- Argument (angle)
- Polar form

## 計算式

$$
\begin{gathered} z = a + bi = r(\cos\theta + i\sin\theta) \\[4pt] r = \sqrt{a^2+b^2},\quad \theta = \operatorname{atan2}(b, a) \\[10pt] z^n = r^n(\cos n\theta + i \sin n\theta) \end{gathered}
$$

## 計算例

### (3 + 4i)(1 − 2i)

- Operation: Multiply z₁ × z₂
- z₁: 3 + 4i
- z₂: 1 - 2i
- **結果: 11 − 2i**
- **Real part: 11**
- **Imaginary part: -2**
- 照合元：3 − 6i + 4i − 8i² = 11 − 2i (hand calculation; Python (3+4j)*(1-2j))

### (3 + 4i) ÷ (1 − 2i)

- Operation: Divide z₁ ÷ z₂
- z₁: 3 + 4i
- z₂: 1 - 2i
- **結果: −1 + 2i**
- **Real part: -1**
- **Imaginary part: 2**
- 照合元：Python (3+4j)/(1-2j) = (-1+2j)

### Polar form of 3 + 4i

- Operation: Convert to polar form
- z₁: 3 + 4i
- **Modulus |z|: 5**
- **Argument (angle): 53.1301023542**
- 照合元：Python decimal: atan2(4, 3) in degrees via Machin-series arctangent (hp.py)

### (1 + i)⁸

- Operation: Power z₁ⁿ
- z₁: 1 + i
- n: 8
- **結果: 16**
- **Real part: 16**
- **Imaginary part: 0**
- 照合元：Python (1+1j)**8 = (16+0j); (√2)⁸ = 16 at angle 8 × 45° = 360°

### Cube roots of 8

- Operation: nth roots of z₁
- z₁: 8
- n: 3
- **結果: 2, −1 + 1.732050808i, −1 − 1.732050808i**
- **Real part: 2**
- **Imaginary part: 0**
- 照合元：2·(cos 120k° + i sin 120k°): Python cmath.rect(2, 2πk/3)

### Polar input 2∠90° (edge: exact zero real part)

- Operation: Convert to polar form
- z₁: 2∠90
- **Real part: 0**
- **Imaginary part: 2**
- **Modulus |z|: 2**
- **Argument (angle): 90**
- 照合元：2(cos 90° + i sin 90°) = 2i

## よくある質問

### How do you multiply complex numbers?

Expand the brackets and replace i² with −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. For (3 + 4i)(1 − 2i) that is (3 + 8) + (−6 + 4)i = 11 − 2i. In polar form the rule is shorter: multiply the moduli and add the angles, so 5∠53.13° × 2.236∠−63.43° = 11.18∠−10.30°.

### How do you divide complex numbers?

Multiply the top and bottom by the conjugate of the divisor, which makes the denominator a real number. (3 + 4i)/(1 − 2i) = (3 + 4i)(1 + 2i)/(1² + 2²) = (3 + 6i + 4i + 8i²)/5 = (−5 + 10i)/5 = −1 + 2i. Division by 0 + 0i is undefined.

### How do you convert a complex number to polar form?

The modulus is r = √(a² + b²) and the argument is θ = atan2(b, a), which picks the correct quadrant. For 3 + 4i, r = √(9 + 16) = 5 and θ ≈ 53.130°, so 3 + 4i = 5∠53.130°. Plain arctan(b/a) gives the wrong angle when the real part is negative: −3 − 4i has an argument of about −126.870°, not 53.130°.

### What is De Moivre's theorem?

For z = r(cos θ + i sin θ) and a whole number n, zⁿ = rⁿ(cos nθ + i sin nθ). So (1 + i)⁸, with r = √2 and θ = 45°, equals (√2)⁸ = 16 at an angle of 8 × 45° = 360°, which is the real number 16. The same idea gives n evenly spaced nth roots: the cube roots of 8 are 2 and −1 ± 1.732051i, 120° apart.

### What is the square root of −1?

The imaginary unit i, defined by i² = −1; −i is the other square root. Every negative number has two imaginary square roots, so √−4 = ±2i, and the principal root, the one with argument 90°, is 2i. Powers of i repeat every four steps: i, −1, −i, 1, then i again.

### 「複素数計算機」の精度はどのくらいですか？

精度は入力値と計算方法の前提に依存します。十進演算には有効数字50桁を使いますが、推定、数値計算手法、元データの精度はそれより低い場合があります。表示の丸め処理でこれらの制約がなくなるわけではありません。 独立した出典の解答と照合した計算例：7。 例えば、「(3 + 4i)(1 − 2i)」は3 − 6i + 4i − 8i² = 11 − 2i (hand calculation; Python (3+4j)*(1-2j))と照合しています。

### この計算方法の出典は何ですか？

Wolfram MathWorld — Complex Number; Wolfram MathWorld — de Moivre's Identity.

## 出典

- [Wolfram MathWorld — Complex Number](https://mathworld.wolfram.com/ComplexNumber.html)
- [Wolfram MathWorld — de Moivre's Identity](https://mathworld.wolfram.com/deMoivresIdentity.html)